The Derivative

Introduction and Summary

Geometric Definition of Derivative

Derivative
-
The instantaneous rate of change of a function
f (x)
at a particular point
x0
, denoted
f'(x0)
or
(x0)
. The formal definition is the following limit:

f'(x0) =

Derivative Function
-
A function
f'(x)
taking on the value of the derivative of
f (x)
at each number
x
.

Difference Quotient
-
For a function
f
, and any two points
x0
and
x
in its domain, the quotient:

Differentiable
-
A function
f
is differentiable at a point (
x0
) if it has a well-defined derivative
at that point. It is simply differentiable if it is differentiable at every point in its
domain.

Secant Line
-
A line through two distinct points on the graph of a function.

Tangent Line
-
A line through a point
(x, f (x))
on a graph that has slope equal to the derivative of the
function
f
at
x
. A limit of secant lines to the graph of a function as two contact
points approach one another.